APU is short for Auxulary Power Unit. Some aircraft it can only be used on the ground. Larger jets need bleed air (air from the engine compressor section) to start their main engines, they use the APU bleed air for this. The APU is started with an electrical motor using the ship's battery. Aircraft that have electrical starters for their main engines, it is better to have electricity supplied by the APU than drain the batteries, or it may not be possible to start with just the battries as there isn't enough charge.
I know in the EMB-120 (30pax turboprop) often you can use the APU to run the packs (presurisation and air conditioning) and turn off the engine bleeds all together to get a little more power on the take off run.
Here's an interesting fact, on the ATR they use a prop lock on one engine and use it as a make shift 'APU'
Hope that answers some of your questions,